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SageMath
E = EllipticCurve("dz1")
E.isogeny_class()
Elliptic curves in class 266805.dz
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 266805.dz1 | 266805dz4 | \([1, -1, 0, -30683509929, -2068731163935122]\) | \(21026497979043461623321/161783881875\) | \(24581429274189918302431875\) | \([2]\) | \(353894400\) | \(4.4645\) | |
| 266805.dz2 | 266805dz2 | \([1, -1, 0, -1918996074, -32278370737145]\) | \(5143681768032498601/14238434358225\) | \(2163386507330358597635496225\) | \([2, 2]\) | \(176947200\) | \(4.1179\) | |
| 266805.dz3 | 266805dz3 | \([1, -1, 0, -1162603899, -57983753690780]\) | \(-1143792273008057401/8897444448004035\) | \(-1351876953902178979748055494835\) | \([2]\) | \(353894400\) | \(4.4645\) | |
| 266805.dz4 | 266805dz1 | \([1, -1, 0, -168488469, -57477454952]\) | \(3481467828171481/2005331497785\) | \(304689897490490131138488585\) | \([2]\) | \(88473600\) | \(3.7714\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 266805.dz have rank \(1\).
Complex multiplication
The elliptic curves in class 266805.dz do not have complex multiplication.Modular form 266805.2.a.dz
sage: E.q_eigenform(10)
Isogeny matrix
sage: E.isogeny_class().matrix()
The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
